169 Pages·2006·0.73 MB·English

COMENIUS UNIVERSITY BRATISLAVA FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF DIDACTIC MATHEMATICS THE CONCEPT OF VARIABLE IN THE PASSAGE FROM THE ARITHMETICAL LANGUAGE TO THE ALGEBRAIC LANGUAGE IN DIFFERENT SEMIOTIC CONTEXTS Doctoral Thesis by ELSA DEL PILAR MALISANI SUPERVISOR: PROF. FILIPPO SPAGNOLO To Carolina, Andrea and Andrés ACKNOWLEDGMENTS My most sincere thanks go to my supervisor, Professor Filippo Spagnolo, for his great availability, for his unconditioned support and for having encouraged and solicited me to write this thesis. I want also to thank Professor Ivan Tren(cid:254)anský for his attention and the precious support he gave me during the realization of the present work. My gratitude goes to the Professors Luis Radford, Vladislav Rosa, Jozef Fulier and Michaela Regecová for having accepted the task of “reporters”. I thank to Nanette Sortino and Giulia Sarullo for having helped me with the translation into English and for having understood the complex problem of my research. All of my gratitude goes to my husband Andrés, to my son Andrea and to my daughter Carolina for their incomparable patience, and for having given me the strength to reach the end of this important journey. A particular thanks to my family of origin, I owe what I am to them. To all of you and many people that I do not mention here for space constraints, I just want to say… Thanks, and God Bless You! 1 INDEX Introduction ……………………………………………………….….. 6 Story of the preceding works .………………………………………………………6 Purpose of the research ……………………………………………………………8 Applications ……………………………………………………………………….. 9 Structure of the thesis………………………………………………………………. 9 Bibliography ……………………………………………………………………….10 Chapter 1. Historical evolution of the algebraic language 12 1.1 Introduction …………………………………………………………………….12 1.2 The symbolism ………………………………………………………………….13 1.3 Methods of resolution of equations …………………………………………….18 1.3.1. Methods of resolution of the equations of first degree ……………...…...18 1.3.1.1. The geometric procedure of Euclid ……………………………18 1.3.1.2. The methods of the false position ……………………………….18 1.3.1.2.1. The methods of the simple false position ……………. 19 1.3.1.2.2. The methods of the double false position ……………. 20 1.3.1.3. The regula infusa ………………………………………………. .22 1.3.2. Methods of resolution of the equations of second degree ….………....... 24 1.3.2.1. The geometric procedure of Euclid .…………………………. 24 1.3.2.2. The procedure of al-Khowârizmî ……………………………… 25 1.3.2.3. The “cut-and-paste geometry” …………………………………. 26 1.3.3. Methods of resolution of the equations of third degree ………………… 27 1.3.3.1. The procedure of al-Khayyam ………………………………… 27 1.3.3.2. The procedure of al-Tusi ………………………………………. 29 1.3.4 Methods of resolution of the indeterminate equations ………………….. 30 1.3.4.1 The procedure of Diophantus ………………………………….. 30 1.3.4.2 The method of pulverization ………………………….………... 30 1.3.4.3 The procedure of Abu Kamil …………………………………… 32 1.3.5. European methods up to 1500 ………………………………………….. 32 1.3.6 Conclusions on the methods of resolutions ………………………………34 1.4 The negative numbers as obstacle. The incomplete numerical field ………….. 36 1.5 Generalization of the problems ……………………………………………….. 38 2 1.6 The variable as “thing that varies” ……………………………………………..39 1.7 Conclusions …………………………………………………………………..... 41 Notes ………………………………………………………………………………. 43 Bibliography ………………………………………………………………………. 45 Chapter 2. The magic square. An experience on the transition between the arithmetical language and the algebraic language 48 2.1. Introduction …………………………………………………………………… 48 2.2. Hypothesis …………………………………………………………………….. 50 2.3. Experimental reports for scholastic level …………………………………….. 50 2.3.1 Middle School ………………………………………………………….... 50 2.3.1.1 The a-didactic situation and its phases ………………………….. 50 2.3.1.2 The a-priori analysis …………………………………………….. 51 2.3.1.3 Quantitative analysis of the data …………………………………52 2.3.1.4 Qualitative analysis ……………………………………………... 54 2.3.1.5 Discussion of the results ………………………………………… 56 2.3.2 High School ……………………………………………………………… 57 2.3.2.1 The a-didactic situation and its phases ………………………….. 57 2.3.2.2 Description of the phase of validation ………………………….. 59 2.3.2.3 The a-priori analysis …………………………………………….. 59 2.3.2.4 Quantitative analysis of the data …………………………………61 2.3.2.5 Qualitative analysis ………………………………………………63 2.3.2.6 Discussion of the results …………………………………………64 2.4 Conclusions ……………………………………………………………………. 65 Bibliography ………………………………………………………………………..66 Appendix N° 1 …………………………………………………………………….. 68 Appendix N° 2 …………………………………………………………………….. 69 Appendix N° 3 …………………………………………………………………….. 70 Appendix N° 4 ……………………………………………………………………. 72 Appendix N° 5 ……………………………………………………………………. 74 Appendix N° 6 ………………………….…………………………………………. 75 3 Chapter 3. The notion of variable in different semiotic contexts 84 3.1. Introduction …………...……………………………………………………… 84 3.2 Methodology of the research……………………………………………………85 3.3 A- priori analysis ……………………………………………………………… 87 3.4 The hypotheses and the a-priori table ………………………………………… 90 3.5 Quantitative analysis ………………………………………………………….. 92 3.5.1 First implicative analysis and comments of the first problem ………….. 93 3.5.2 Falsification of H ………………………………………………………. 95 1 3.5.3 Profile of the pupils ………………………………………………………95 3.5.4 The hierarchical tree …………………………………………………….. 96 3.5.5 The factorial analysis by S.P.S.S. ……………………………………….. 97 3.5.6 Second implicative analysis and comments of the first problem ……….. 98 3.5.7 Falsification of H ……………………………………………………… 99 2 3.5.8 Third implicative analysis and comments of the second problem ……… 100 3.5.9 Falsification of H ………………………………………………………. 101 3 3.5.10 Fourth implicative analysis and comments of the fourth problem ……. 101 3.5.11 Fifth implicative analysis and comparison between the first and fourth problem ……………………………………………………………….. 103 3.5.12 Falsification of H ………………………………………………………104 4 3.6 Conclusions ……………………………………………………………………. 104 Notes ………………………………………………………………………………. 106 Bibliography ……………………………………………………………………….107 Appendix N° 1 …………………………………………………………………….. 110 Appendix N° 2 ……………………………………………………………………...111 Appendix N° 3 …………………………………………………………………….. 114 Appendix N° 4 …………………………………………………………………….. 116 Appendix N° 5 ……………………………………………………………………..116 Chapter 4. The variable between unknown and “thing that varies”. Some aspects of the symbolic language 117 4.1. Introduction ………………………………………………….……………….. 117 4.2. Methodology of the research ……... …………………………………………. 118 4 4.3 Analysis of the protocols of the first problem …………………………………118 4.3.1 First pair: Serena and Graziela …………………………………..…….. 118 4.3.1.1 Types of language ……………………………………………… 118 4.3.1.2 Resolutive procedure ………………………………………….. 119 4.3.1.3 Comments ……………………………………………………… 119 4.3.2 Second pair: Vita and Alessandra ………………………………………. 120 4.3.2.1 Types of language ……………………………………………… 120 4.3.2.2 Resolutive procedure ……………………………………………120 4.3.2.3 Comments ………………………………………………………122 4.4 Analysis of the protocols of the second problem ………………………………124 4.4.1 First pair: Serena and Graziela ………………………………………… 124 4.4.1.1 Resolutive procedure ………………………………………….. 124 4.4.1.2 Comments ………………………………………………………125 4.4.2 Second pair: Vita and Alessandra ………………………………………. 125 4.4.2.1 Resolutive procedure ………………………………………….. 125 4.4.2.2 Comments ……………………………………………………... 127 4.5 Final conclusions …………………………………………………………….. 129 Notes ………………………………………………………………………………132 Bibliography ………………………………………………………………………132 Appendix N° 1 ……………………………………………………………………..134 Appendix N° 2 ……………………………………………………………………. 135 Appendix N° 3 …………………………………………………………………… 139 Chapter 5. Final conclusions 160 Bibliography ……………………………………………………………………… 167 5 INTRODUCTION STORY OF THE PRECEDING WORKS The aim of the experimental research effected in Malisani (1990, 1992) was to study the cognitive performance of the students between the ages of 14-15 in the assignment of resolution of algebraic and geometric problems. We wanted to know more specifically how the different kind of logical structure of a problem affect the resolving performance (types of solutions, steps of the resolving algorithm and errors); and if it is verified that the isomorphism of logical structures in the algebraic and geometric contexts does not implicate isomorphism in the performance of the students. The problems belonging to the algebraic context are concerned with the resolution of equations of first degree with one unknown, of the type: y = k . (x - k ) [1] for a 1 2 determined value of x or y, being k and k positive constants and such that k < x. In 1 2 2 this type of equation the variables x and y and the constants k and k can represent any 1 2 elements, therefore, they have only a formal significance. The geometric problems consider, instead, the application of the “theorem of the sum of the interior angles of a convex polygon” that has equation: s = 180°. (n - 2) [2], in which every variable and every constant represent determined geometric objects or relations among these objects. For example: s is the sum of the interior angles of a convex polygon, 180° is the sum of the interior angles of a triangle, n is the number of sides of a polygon, n (cid:177) 2 is the number of triangles that are determined in the polygon tracing the diagonals from a vertex to the others. In this case the variables and the constants have a geometric significance. We observe that the equations [1] and [2] are isomorphic with regard to their logical structure, because if we fix a variable (for example: y and s) they require the same steps for their resolution. These equations are of arithmetical kind, using the terminology of Gallardo and Rojano (1988), because to resolve them it is necessary to manipulate only the numerical values of the equation (actions in the arithmetic context) and not the quantities to find or unknowns. From the results obtained we deduce that the geometric significance of a problem: 6 (cid:216) would affect partially the achievement of correct answers, only in those problems that introduce greater logical difficulty (7 or more different steps). (cid:216) influences positively the economy of steps of the resolution. (cid:216) affects the number and the kind of errors made by the pupils. Therefore, the intuitive support that the geometric problems offer and a good comprehension by the pupils of the conceptual relations between the elements that compose the equation of the theorem favour the saving of steps in the resolution and decrease the number of errors, above all the errors of calculation. That is, the saving of steps in the resolution does not always implicate a greater quantity of errors, contrary to what is usually supposed. At that time the formulated conclusions affirmed that the resolution of problems that involves equations requires something more than the domain of certain operations (arithmetical and algebraic); the subjects must have the necessary conceptual knowledge to understand and to represent conveniently the information of the problem. The individualization and the diagnosis of the errors effected in Malisani (1990) and Malisani (1992) led us to deepen the principal works of research carried out in these last decades on the cognitive processes associated with the learning of algebra (Matz, 1982; Kieran & Filloy, 1989; Kieran, 1991; Gallardo & Rojano, 1988; Lee & Wheeler, 1989; Chiappini and Lemut, 1991; Herscovics & Linchevski, 1991). These studies deal with matters concerning the difficulties and obstacles that the beginner students of algebra meet, regarding the conceptual changes necessary in the transition from the arithmetic thought to the algebraic thought. These changes refer especially to the concept of equality, the conventions of notation and the interpretation of the concept of variable. We also examined the results of some researches on the interpretation and simplification of algebraic expressions and the resolution of equations and algebraic problems (Malisani, 1993). Successively we carried out a research on the individualization, diagnosis and classification of errors in the resolution of algebraic and geometric problems that involve arithmetical equations of first degree (Malisani, 1993). Even if the resolving procedure of the algebraic and geometric problems is isomorphic two by two, the results we obtained point out that the students do not make the same types of errors. For example, the percentages of errors related to the use of the equal sign and to the transport of terms from a member to the other of the equation are lower in the geometric 7 context. On the other hand, the percentage of errors concerning the formulation of an answer consistent with the meaning of the variables that represent the results is lower in the algebraic context. Several experimental studies (Harper, 1987; Sfard 1992) seem to confirm that some difficulties of the students can be grouped around some obstacles met in history (Cfr. Arzarello, pp. 7-8). The elements that allow to identify these obstacles have to be searched in the analysis of the resistances emerged in the historical development and in the debates that have overcome them. But history alone is not sufficient; the historical epistemological analysis must be completed by a study of the grounding of mathematics (Spagnolo, 1995, pp. 18-19). If we consider this point of view, it could be useful to take into consideration the history of the algebraic thought that leads us to go over the steps of the construction of the algebraic language. The historical analysis effected in Malisani (1996, 1999) shows that for many centuries algebra stayed behind in comparison with geometry and that the construction of the symbolic language was too slow and difficult. The lack of an adequate algebraic language conditioned the evolution of the resolutive procedures. The ancient mathematicians often explained these procedures through their application to some examples. They used other languages: natural, arithmetical and geometric. PURPOSE OF THE RESEARCH To deepen the conclusions previously expressed a new research is proposed. It is founded on the necessity of studying and analyzing the obstacles that the students meet in building up and assimilating certain concepts, in the passage from the arithmetical thought to the algebraic thought. From some effected studies (Matz, 1982; Wagner, 1981, 1983) it emerges that the point of critical transition between the two kinds of thought is the introduction of the concept of variable. This notion could take on a plurality of conceptions: generalized number (2+4 = 4+2 is generalized with a+b = b+a); unknown (resolution of equations); (cid:179)something that varies(cid:180) (relation among quantities, functional aspect); entirely arbitrary sign (study of the structures); register of memory (in computer science) (Usiskin, 1988). The study of the various aspects that this concept can take constitutes a very wide field of research and requires different confirmations, provided by historical-epistemological 8 and experimental investigation and by setting up the didactical situations built ad-hoc. Therefore it is necessary to circumscribe the dominion of survey. The aim of this research is to study some characteristics of the period of transition from the arithmetical language to the algebraic language. We want to analyze if the different conceptions of variable are evoked by the students in the resolution of problems and if the notion of variable in its double aspect – unknown and relational-functional – represents an obstacle for the pupil. APPLICATIONS This research is set as a contribution to Mathematics Education, particularly, to the studies that are being carried out within the GRIM, on the epistemological and didactical obstacles concerning the passage from the arithmetic language to the algebraic language. This experimental study will supply us some necessary tools to analyze in details whether the concept of variable, in its different aspects, represents an epistemological obstacle or an obstacle of didactical origin. We could also determine how the semiotic context influences the conceptions of variable from the pupil’s point of view. We could study more specifically the interaction of other contexts –natural language, geometric language, perceptive schemes, etc.– with the operating of the pupils in a strictly algebraic context. Moreover, it will be possible to draw some tools to set up appropriate a-didactical situations and to get at a more deep comprehension of the communicative processes. From a general point of view, this research can help to clarify matters concerning the representations of the arithmetical and algebraic knowledge and the operating in the resolution of problems from the pupil’s point of view. STRUCTURE OF THE THESIS The thesis is composed by five chapters. The first one is about history and introduces the construction of the algebraic language and the evolution of the methods and of the strategies of resolution of equations in the periods that preceded the formalization. The second chapter has the purpose to study some aspects of the period of transition from the arithmetical language to the algebraic language. We want to analyze if the pupils evoke the different conceptions of variable in the resolution of problems and if 9

My gratitude goes to the Professors Luis Radford, Vladislav Rosa, Jozef Fulier and 120 e 71m .. equations (Cfr. Charbonneau & Radford, pp. 2).

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